Here There Be (Dusty) Monsters: High Redshift AGN are Dustier Than Their Hosts
JWST spectroscopy has discovered a population of $z \gtrsim 3.5$ galaxies with broad Balmer emission lines, and narrow forbidden lines, that are consistent with hosting active galactic nuclei (AGN). Many of these systems, now known as ``little red dots" (LRDs), are compact and have unique colors that are very red in the optical/near-infrared and blue in the ultraviolet. The relative contribution of galaxy starlight and AGN to these systems remains uncertain, especially for the galaxies with unusual blue+red spectral energy distributions. In this work, we use Balmer decrements to measure the independent dust attenuation of the broad and narrow emission-line components of a sample of 29 broad-line AGN identified from three public JWST spectroscopy surveys: CEERS, JADES, and RUBIES. Stacking the narrow components from the spectra of 25 sources with broad H$\rmα$ and no broad H$\rmβ$ results in a median narrow H$\rmα$/H$\rmβ$ = $2.47^{+0.05}_{-0.05}$ (consistent with $A_{v} = 0$) and broad H$\rmα$/H$\rmβ$ $> 8.85$ ($A_{v} > 3.63$). The narrow and broad Balmer decrements imply little-to-no attenuation of the narrow emission lines, which are consistent with being powered by star formation and located on larger physical scales. Meanwhile, the lower limit in broad H$\rmα$/H$\rmβ$ decrement, with broad H$\rmβ$ undetected in the stacked spectrum of 25 broad-H$\rmα$ AGN, implies significant dust attenuation of the broad-line emitting region that is presumably associated with the central AGN. Our results indicate that these systems, on average, are consistent with heavily dust-attenuated AGN powering the red parts of their SED while their blue UV emission is powered by unattenuated star formation in the host galaxy.